Angle estimation method for MIMO radar with element failure based on factor matrix prior

Abstract

The present invention discloses an angle estimation method for MIMO radar with element failure based on factor matrix prior. The method comprises the following steps: performing matched filtering on a received signal of a bistatic MIMO radar with element failure to obtain an output signal #imgabs0# of a virtual array and calculating a virtual array covariance matrix #imgabs1#; constructing a fourth-order covariance tensor #imgabs2# and expressing it as a tensor CANDECOMP / PARAFAC decomposition model; establishing a tensor filling model with factor matrix prior constraints; converting the tensor filling model into an unconstrained augmented Lagrangian function form; iteratively solving the augmented Lagrangian function using an ADMM algorithm, and obtaining a factor matrix U at the end of the iteration. (1) , U (2) and the diagonal matrix Δ (3) , Δ (4) The complete covariance tensor #imgabs3# is constructed and then restored to the complete covariance matrix R through a symmetric Hermitian expansion. Finally, the ESPRIT algorithm is used to estimate the target angle. This method can effectively recover missing data for multiple slices in the fourth-order covariance tensor, improving the angle estimation performance of MIMO radars under array element failure.

Description

Technical Field

[0001] The present invention relates to MIMO radar angle estimation, in particular to an element failure MIMO radar angle estimation method based on factor matrix prior. Background Art

[0002] Different from the mechanism of traditional phased array radar, Multiple Input Multiple Output (MIMO) radar uses multiple transmitting antennas to simultaneously transmit multiple orthogonal signal waveforms, and multiple receiving antennas to receive target echo signals and separate the target information of each channel through matched filters, thereby improving the maximum number of identifiable targets, anti-interference ability and parameter estimation accuracy.

[0003] Target azimuth estimation is a major research topic in array signal processing. Bistatic MIMO radars employ a bistatic array configuration with separate transmitters and receivers, enabling simultaneous estimation of the target's direction of departure (DOD) and direction of arrival (DOA) at the receiver. Currently, algorithms such as MUSIC, Capon, and ESPRIT have been proposed to address the target angle estimation problem in bistatic MIMO radars, leveraging the virtual array covariance matrix. The ESPRIT algorithm has garnered widespread attention due to its ability to avoid two-dimensional spectral peak searching and significantly reduce computational complexity. In practical applications, due to harsh operating environments and component aging, elements in the transmit and receive arrays of long-term MIMO radars may fail. This disrupts the geometric structure of the radar array, resulting in a large number of failed elements in the virtual array and, consequently, missing data in entire rows and columns in the virtual array covariance matrix, inevitably degrading radar angle estimation performance. Therefore, effectively compensating for the missing data caused by failed elements is crucial for MIMO radar angle estimation.

[0004] In the paper "Joint Sensor Failure Detection and Corrupted Covariance Matrix Recovery in Bistatic MIMO Radar With Impaired Arrays" (IEEE Sensors Journal, 2019, 19(14):5834-5842), Chen et al. proposed a MIMO radar angle estimation algorithm based on block Hankel matrix reconstruction. This method constructs the covariance matrix as a quadruple Hankel matrix so that each row and column in the reconstructed matrix contains non-zero elements. Then, the matrix completion algorithm (MC) is used to recover the missing data. However, since the Hankel operation will increase the matrix dimension, this method has high computational complexity and long operation time. Most existing MIMO radar element failure angle estimation methods recover the faulty array data in a matrix framework, but the matrix form cannot well reflect the multidimensional constraint relationship between the data. Therefore, the performance of such methods in recovering the missing data of the failed array element needs to be further improved. In order to utilize the multidimensional structure of tensor data, the virtual array covariance data matrix of the MIMO radar can be constructed into a fourth-order covariance tensor. However, due to array element failure, some slices are completely missing in the fourth-order covariance tensor of MIMO radar. In this case, the existing tensor filling algorithm cannot effectively recover this structural missing data. In the paper "Hankel Matrix Nuclear Norm Regularized Tensor Completion for N-dimensional Exponential Signals" (IEEE Transactions on Signal Processing, 2017, 65 (14): 3702-3717), Ying et al. proposed a tensor filling algorithm based on Hankel matrix nuclear norm regularization (HMRTC). This method combines the low CANDECOMP / PARAFAC (CP) rank of the tensor and the Vandermonde structure of the factor matrix, imposes a low rank constraint on the Hankel matrix constructed by the column vectors of each factor matrix, and can reconstruct a multidimensional complete signal from a small amount of collected data, thereby recovering the structural missing data in the tensor. Since the factor matrix of the MIMO radar covariance tensor after CP decomposition has a Vandermonde structure, the HMRTC algorithm is applicable to the reconstruction of missing data of failed array elements in MIMO radar.However, the HMRTC algorithm fails to consider the relationships between the factor matrices after tensor decomposition when recovering missing data, resulting in suboptimal recovery performance for failed element data. In MIMO radars with element failures, leveraging the multidimensional structural characteristics of the echo signal and prior information about the tensor factor matrix to improve the accuracy of failed element data recovery is crucial for enhancing the radar's resilience to element failures. Summary of the Invention

[0005] Purpose of the invention: The purpose of the present invention is to provide an angle estimation method for MIMO radar with array element failure based on factor matrix prior, so as to quickly recover the missing data in the MIMO radar under array element failure, thereby improving the angle estimation accuracy under array element failure.

[0006] Technical solution: The present invention provides a method for estimating the angle of an element failure MIMO radar based on a factor matrix prior, comprising the following steps:

[0007] (1) Matched filtering is performed on the received signal of the bistatic MIMO radar with array element failure to obtain the output signal of the virtual array And calculate the virtual array covariance matrix

[0008] (1.1) Both the transmit array and receive array of a bistatic MIMO radar are uniform linear arrays, consisting of M transmit elements and N receive elements, respectively. When an element fails, the echo signal collected by the bistatic MIMO radar is subjected to matched filtering. After pulse accumulation, the resulting echo signal matrix output by MN virtual elements is:

[0009] in, is the virtual array echo signal matrix under Q snapshots; and are the manifold matrices of the transmitting array and receiving array when there are failed array elements; is the Gaussian white noise matrix; S∈ P×Q is the target coefficient matrix, where P is the independent far-field target; ⊙ represents the Khatri-Rao product;

[0010] (1.2)Ω T and Ω R They are the position sets of failed transmitting array elements and failed receiving array elements respectively. When the pth t ∈Ω T When an array element fails, its manifold matrix The first p t The rows are all zero; when the pth row in the receiving array r ∈Ω RWhen an array element fails, the manifold matrix The first p r All rows are zero; the maximum likelihood estimate of the virtual array covariance matrix under the Q pulse period is:

[0011] in,(·) H represents the conjugate transpose, R s =SS H / Q=diag(ρ) represents the signal covariance matrix, diag(ρ) represents the diagonal matrix generated by the vector ρ, ρ=[ρ1,ρ2,…,ρ P ],ρ p ,p=1,2,…,P, represents the reflection coefficient of the pth target, represents the noise covariance matrix.

[0012] (2) Using the virtual array covariance matrix Constructing a fourth-order covariance tensor The covariance tensor Represented as a tensor CANDECOMP / PARAFAC (CP) decomposition model.

[0013] (2.1) Dual-base MIMO radar virtual array covariance matrix Can be rearranged into a fourth-order covariance tensor in,(·) * indicates conjugation; Represents Tucker operation.

[0014] (2.2) Re-adjust the factor matrix Make all elements in g equal to 1. Then eliminate g to get the simplified expression of the covariance tensor

[0015] (2.3) Due to the perturbation of the permutation and scaling effects of the CP decomposition, the covariance tensor The factor matrix is expressed as and in and represents four factor matrices; represents the permutation matrix; Λ1, Λ2, Λ3 and Λ4 are P×P real-valued diagonal matrices, and the diagonal elements correspond to the scaling coefficients; N1, N2, N3 and N4 represent the fitting errors.

[0016] (2.4) Since the transmit and receive steering matrices A of the MIMO radar t 、A r are all Vandermonde structures, and the factor matrix and Each column vector of A is a different constant t and A r The corresponding column vectors of are scaled respectively, so when the fitting error term is ignored, the matrix and It also has a Vandermonde structure, that is, the factor matrix U (n) ,n=1,…,4 has a Vandermonde structure.

[0017] (2.5) Ignoring the influence of the fitting error term, the relationship between the factor matrices is: and The diagonal matrix Diagonal matrix

[0018] (3) By utilizing the Vandermonde structure characteristics of the factor matrix, the mutual relationship between the factor matrices and the low CP rank of the tensor, a tensor filling model with prior constraints on the factor matrix is established.

[0019] In order to effectively recover the missing data of failed array elements, a tensor filling model for recovering missing data of failed array elements is established by utilizing the Vandermonde structure characteristics of the factor matrix, the mutual relationship between the factor matrices, and the low CP rank of the tensor:

[0020]

[0021] Among them, M (n) ,n=1,2,3,4, represents the auxiliary matrix, Ω represents the incomplete covariance tensor The set of known non-zero elements in ; represents the projection operator onto the set Ω; ||·|| * represents the nuclear norm; ||·|| F represents the Frobenius norm; λ represents the regularization parameter; Represents a block Hankel matrix operation, defined as the matrix Transform to block Hankel matrix operation, where u p represents the p-th column of U, p=1,2,…,P; S1+S2=S+1; Represents the Hankel operation, defined as the vector Transformed into Hankel matrix operation, where S1+S2=S+1.

[0022] (4) In order to solve the constrained tensor filling model, the tensor filling model in step (3) is converted into an unconstrained augmented Lagrangian function form.

[0023] The optimization model in step (3) is expressed as an unconstrained augmented Lagrangian function:

[0024]

[0025] For simplicity, we define and β>0 indicates penalty coefficient; D (n) represents the Lagrange multiplier matrix, n=1, 2, 3, 4; <·> represents the inner product.

[0026] (5) Use the ADMM algorithm to iteratively solve the augmented Lagrangian function in step (4), and obtain the factor matrix U at the end of the iteration (1) ,U (2) and the diagonal matrix Δ (3) ,Δ (4) .

[0027] The optimization problem in step (4) is iteratively solved using the Alternating Direction Method of Multipliers (ADMM) algorithm, where the variables are updated at the kth iteration. and The steps are as follows:

[0028] (5.1) In the kth iteration, by solving the optimization problem To update the variable This sub-problem is transformed into solving U (1) and U (2) Convex optimization problem:

[0029]

[0030] Where, (·) T represents transpose; R (n) and Tensors and The solution of the convex optimization problem (3) satisfies:

[0031]

[0032] in, express The inverse transform of U (n) The closed-form solution of each row is used to obtain U (n) The solution of and W i =diag(Ω(n) (i,:)), i∈{1,2,…,I n}; When n=1, I1=M; when n=2, I2=N; then formula (4) can be transformed into:

[0033]

[0034] Where U (n) (i,:) represents the matrix U (n) The i-th row; C(i,:) represents the i-th row in the matrix C. Therefore, the row vector U (n) The closed-form solution of (i,:) is expressed as:

[0035]

[0036] Then, by solving U (n) Each line Get the factor matrix U (n) =[(U (n) (i,:)) T …(U (n) (I n ,:)) T ] T , n=1,2.

[0037] (5.2) Regarding the solution of Δ={Δ (3) ,Δ (4) The convex optimization problem of} is expressed as:

[0038]

[0039] Similarly, the solution to this convex optimization problem satisfies:

[0040]

[0041] Formula (8) can be further expressed as:

[0042]

[0043] Where, To solve the diagonal matrix of the i-th system of equations, n=3,4,i=1,2,…,I n And when n=3, I3=M, when n=4, I4=N; therefore The closed-form solution can be expressed as:

[0044]

[0045] in, Represents the generalized inverse; To reduce the estimation error, Take the average as Δ(n) Solution:

[0046]

[0047] (5.3) About variables The optimization problem is expressed as:

[0048]

[0049]

[0050] Using the SVT algorithm to solve the matrix nuclear norm minimization problem, the solution to the above optimization problem is:

[0051]

[0052]

[0053] in, is the singular value soft threshold shrinkage operator, 1 / β k is the threshold.

[0054] (5.4) Variables The update expression is:

[0055]

[0056]

[0057] (5.5) The expression for updating β is β k+1 =ρβ k .

[0058] (6) According to the factor matrix U (1) ,U (2) and the diagonal matrix Δ (3) ,Δ (4) , using Tucker operations to construct the complete covariance tensor Then it is restored to the complete covariance matrix R through symmetric Hermitian expansion, and finally the ESPRIT algorithm is used to estimate the target angle.

[0059] The factor matrix U obtained according to step (5) (1) ,U (2) and the diagonal matrix Δ (3) ,Δ (4) , using Tucker operations to construct the complete covariance tensor and will The covariance matrix is expanded through symmetric Hermitian expansion, and the target angle is finally estimated using the ESPRIT algorithm.

[0060] A computer storage medium stores a computer program, which, when executed by a processor, implements the above-mentioned element failure MIMO radar angle estimation method based on factor matrix prior.

[0061] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, the method for estimating the angle of an element failure MIMO radar based on a factor matrix prior is implemented.

[0062] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0063] 1. The virtual array covariance matrix of a bistatic MIMO radar with element failure has entire rows and columns of missing data. To utilize the multidimensional structure of the echo data to recover the missing data of the failed element, the present invention rearranges the virtual array covariance matrix into a covariance tensor. However, the covariance tensor has multiple slices of data completely missing, and traditional tensor filling algorithms cannot reconstruct a covariance tensor with missing slices. The present invention proposes a tensor filling algorithm based on a factor matrix prior that can effectively recover multiple missing slices of data in the fourth-order covariance tensor, improving the angle estimation performance of MIMO radar with element failure.

[0064] 2. The present invention exploits the multidimensional structural characteristics of the fourth-order covariance tensor of the MIMO radar, the Vandermonde structural characteristics of the factor matrix, and the relationship between the factor matrices. It can more accurately recover the missing data in the covariance tensor, thereby better compensating for the angle estimation performance loss caused by array element failure.

[0065] 3. The present invention can effectively solve the problem of data recovery when the entire slice is missing in the MIMO radar covariance tensor, and has good accuracy in recovering missing data from failed array elements and high real-time performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 A flowchart of the steps of the method of the present invention;

[0067] Figure 2 is the target angle estimation constellation diagram of the bistatic MIMO radar; where, Figure 2 (a) When the array element fails, the ESPRIT algorithm is used. Figure 2 (b) When the array element fails, the second existing technology is used. Figure 2 (c) When the array element fails, the method of the present invention is used;

[0068] Figure 3 The figure is a graph showing the convergence change of the method of the present invention and the second prior art;

[0069] Figure 4 This is a curve diagram of target angle estimation RMSE changing with signal-to-noise ratio;

[0070] Figure 5 This is a graph showing the change of RMSE of target angle estimation with the number of snapshots;

[0071] Figure 6 This is a curve diagram showing the change of angle estimation RMSE with the number of receiving faulty array elements when the number of failed transmitting array elements is 1;

[0072] Figure 7 This is a curve diagram of the change of RMSE with the number of receiving faulty array elements when the number of failed transmitting array elements is 2. DETAILED DESCRIPTION

[0073] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0074] like Figure 1 As shown in FIG, the method for estimating the angle of a MIMO radar with element failure based on a factor matrix prior includes the following steps:

[0075] Step 1: Perform matched filtering on the received signal of the bistatic MIMO radar with element failure to obtain the output signal of the virtual array And calculate the virtual array covariance matrix

[0076] The transmitting array of the bistatic MIMO radar has M transmitting elements, and the element spacing is d t The receiving array has N receiving elements, and the element spacing is d r , and they are all uniform linear arrays (ULA). Assume that there are P incoherent targets in the far field, and the distance angle of deflection (DOD) of the pth (p=1,2,…,P) target is θ p , the angle of arrival (DOA) is Each transmitting array element simultaneously transmits mutually orthogonal coded signals W = [w1, w2, ..., w M ] T (i.e. (1 / K)WW H =I M×M ),in, represents the baseband coding signal of the mth (m=1,2,…,M) transmitting array element in each pulse period, K represents the number of phase codes in each pulse period, (·) H Each element in the receiving array receives the target reflected echo, and the received signal X at the qth (q=1,2,…,Q) pulse period is q for:

[0077]

[0078] Where, in Steering vector for the receiving array; in, Steering vector for the transmitting array; Represented by vector s q The diagonal matrix formed by ρ p represents the reflection coefficient of the pth target, f dp represents the Doppler frequency of the pth target, f s is the pulse repetition frequency, (·) T Indicates transpose operation; is the noise matrix.

[0079] By using the mutual orthogonality of the transmitted waveforms, (1 / K)W H The output after matched filtering obtained by multiplying formula (1) on the right is:

[0080]

[0081] Among them, Y q =X q W H / K; Let y q =vec(Y q ), after the echo signal of Q pulse periods is matched filtered, the output of the virtual array is:

[0082] Y=[y1,y2,…,y Q ]=(A r ⊙A t )S+Z (3)

[0083] Where, Then the virtual array covariance matrix can be expressed as:

[0084] R=YY H / Q=(A r ⊙A t )R s (A r ⊙A t ) H +R Z (4)

[0085] Where, is the signal covariance matrix; is the noise covariance matrix. For incoherent targets, R s It can be expressed as a diagonal matrix, namely R s=diag([ρ1,ρ2,…,ρ P ]).

[0086] Assume Ω T and Ω R are the position sets of the faulty transmitting and receiving array elements, respectively, and the vector and Indicates the position of the normal array elements in the transmit array and receive array, where the element is 0 at the index of the faulty array element and 1 at the rest of the positions. Therefore, the faulty transmit and receive array steering matrices can be expressed as

[0087]

[0088]

[0089] The virtual array element output data corresponding to the failed array element is set to zero, and the output data matrix of the bistatic MIMO radar virtual array under array element failure can be expressed as:

[0090]

[0091] Where, is the noise matrix after the output data of the virtual array element corresponding to the failed array element is set to zero.

[0092] Then the virtual array covariance matrix of the bistatic MIMO radar under element failure can be expressed as:

[0093]

[0094] Where, is the noise covariance matrix.

[0095] Step 2: Using the virtual array covariance matrix Constructing a fourth-order covariance tensor The covariance tensor Represented as a tensor CANDECOMP / PARAFAC (CP) decomposition model.

[0096] The covariance matrix R of the bistatic MIMO radar can be rearranged into a fourth-order covariance tensor It can be expressed as:

[0097]

[0098] Where, is a diagonal core tensor whose (p,p,p,p)th element is equal to ρ p , the rest are zero; is the noise covariance tensor.

[0099] Similarly, the covariance matrix of the bistatic MIMO radar under element failure is Can be rearranged into a fourth-order covariance tensor It can be expressed as:

[0100]

[0101] Where, is the noise covariance tensor under array element failure. Analyzing formula (11), we can see that the covariance tensor The missing slice data in can be expressed as:

[0102]

[0103]

[0104] From (12) and (13), we can see that in MIMO radar, failed array elements cause some slices in the fourth-order covariance tensor to be completely missing. Different from the random missing data problem in most tensor completion methods, we call these missing data forms in the covariance tensor structural missing.

[0105] Ignoring the noise term in (10), the covariance tensor The CP decomposition form can be expressed as:

[0106]

[0107] in, The scaling factor matrix can be used to make all elements in g become 1. At this time, g can be eliminated to obtain a simplified expression of the covariance tensor:

[0108]

[0109] The transmit and receive steering matrices of MIMO radar have the characteristics of Vandermonde structure, and the factor matrix and Each column vector of A is a different constant t and A r The corresponding column vector of is scaled, so the factor matrix and It also has a Vandermonde structure.

[0110] Taking into account the permutation and scaling effects of CP decomposition, the covariance tensor The factor matrix of can be estimated as:

[0111]

[0112]

[0113]

[0114]

[0115] Where, and represents the factor matrix; represents the permutation matrix; Λ1, Λ2, Λ3, and Λ4 are P×P real-valued diagonal matrices, with the diagonal elements corresponding to the scaling coefficients; N1, N2, N3, and N4 represent the fitting errors. Ignoring the fitting error terms in Equation (16), we can obtain from (16a):

[0116]

[0117] Substituting (17) into (16c) yields:

[0118]

[0119] in, is a diagonal matrix. Similarly, U (2) 、U (4) The relationship between can be expressed as:

[0120]

[0121] Step 3: Utilizing the Vandermonde structure characteristics of the factor matrix, the mutual relationship between the factor matrices and the low CP rank of the tensor, a tensor filling model with prior constraints on the factor matrix is established.

[0122] In order to effectively recover the missing data of failed array elements, a tensor filling model for recovering missing data of failed array elements is established by utilizing the Vandermonde structure characteristics of the factor matrix, the mutual relationship between the factor matrices, and the low CP rank of the tensor:

[0123]

[0124] Where Ω represents the incomplete covariance tensor The set of known non-zero elements in ; represents the projection operator onto the set Ω; ||·|| * represents the nuclear norm; ||·|| F represents the Frobenius norm; λ represents the regularization parameter; Represents a block Hankel matrix operation, defined as the matrix Transform to block Hankel matrix Among them, S1+S2=S+1; Represents the Hankel operation, defined as the vector Transformed into Hankel matrix Among them, S1+S2=S+1, which can be expressed as:

[0125]

[0126] at the same time, for The inverse operation is specifically expressed as:

[0127]

[0128] in, for The inverse operation can be expressed as:

[0129]

[0130] according to and The definition of , we can get:

[0131]

[0132] Step 4: To solve the constrained tensor filling model, the tensor filling model in step (3) is converted into an unconstrained augmented Lagrangian function form.

[0133] The optimization model of formula (20) is expressed as an augmented Lagrangian function:

[0134]

[0135] For simplicity, we define and β>0 indicates penalty coefficient; D (n) (n=1, 2, 3, 4) represents the Lagrange multiplier matrix; <·> represents the inner product.

[0136] Step 5: Use the ADMM algorithm to iteratively solve the augmented Lagrangian function in step 4, and obtain the factor matrix U at the end of the iteration (1) ,U (2) and the diagonal matrix Δ (3) ,Δ (4) .

[0137] The ADMM (Alternating Direction Method of Multipliers) algorithm is used to solve Equation (25). When iteratively optimizing a variable, the latest values of other variables are fixed. and The steps are as follows:

[0138] Step 5.1: In the kth iteration, solve the optimization problem To update the variable , transform this sub-problem into the problem about U (n) Convex optimization problem:

[0139]

[0140] in, R (n) and Tensors and The solution of this convex optimization problem satisfies:

[0141]

[0142] Where, express The inverse transform of U (n) The closed-form solution of each row is used to obtain U (n) The solution of . Definition and W i =diag(Ω (n) (i,:)), i∈{1,2,…,I n}, and when n = 1, I1 = M, when n = 2, I2 = N. Then formula (27) can be transformed into:

[0143]

[0144] Where U (n) (i,:) represents the matrix U (n) The i-th row; C(i,:) represents the i-th row in the matrix C. Therefore, the row vector U (n) The closed-form solution of (i,:) can be expressed as:

[0145]

[0146] Then, we can solve U (n) Each line Get the factor matrix U (n) =[(U (n) (i,:)) T …(U (n) (I n ,:)) T ] T , n=1,2.

[0147] Step 5.2: The convex optimization problem with respect to Δ is expressed as:

[0148]

[0149] Similarly, the solution to this convex optimization problem satisfies:

[0150]

[0151] Formula (8) can be further expressed as:

[0152]

[0153] in, To solve the diagonal matrix of the i-th system of equations, n = 3, 4; i = 1, 2, ..., I n , and when n=3, I3=M; when n=4, I4=N. Therefore The closed-form solution can be expressed as:

[0154]

[0155] in, In order to reduce the estimation error, Take the average as Δ (n) Solution:

[0156]

[0157] Step 5.3: About variables The optimization problem is expressed as:

[0158]

[0159]

[0160] The SVT algorithm is used to solve the matrix nuclear norm minimization problem. The closed-form solution of the above optimization problem is:

[0161]

[0162]

[0163] in, is the singular value soft threshold shrinkage operator, 1 / β k is the threshold.

[0164] Step 5.4: Variable D (n) The update expression is:

[0165]

[0166]

[0167] Step 5.5: Update the expression of β to βk+1 =ρβ k .

[0168] Step 6: According to the factor matrix U (1) ,U (2) and the diagonal matrix Δ (3) ,Δ (4) , using Tucker operations to construct the complete covariance tensor Then it is restored to the complete covariance matrix R through symmetric Hermitian expansion, and finally the ESPRIT algorithm is used to estimate the target angle.

[0169] The factor matrix U obtained from step 5 (1) ,U (2) and the diagonal matrix Δ (3) ,Δ (4) , using Tucker operations to construct the complete covariance tensor and will The covariance matrix is expanded through symmetric Hermitian expansion, and the target angle is finally estimated using the ESPRIT algorithm.

[0170] The technical effect of the present invention can be illustrated by the following simulation experiments. The DOA estimation performance of the method of the present invention is compared with that of the prior art one (Jinli Chen, Tingxiao Zhang, Jiaqiang Li, et al. Joint Sensor Failure Detection and Corrupted Covariance Matrix Recovery in Bistatic MIMO Radar With Impaired Arrays [J]. IEEE Sensors Journal, 2019, 19 (14): 5834-5842) and the prior art two (J. Ying et al. Hankel Matrix Nuclear Norm Regularized Tensor Completion for N-dimensional Exponential Signals, IEEE Transactions on Signal Processing, 2017, 65 (14): 3702-3717), and the angle estimation performance of directly using the ESPRIT algorithm when the array element is normal is used as a reference. It should be noted that in order to fairly compare the performance of various methods for recovering the missing data of failed array elements, the ESPRIT algorithm is used to estimate the target angle from the complete data reconstructed by different methods. Assume that the number of transmitting and receiving array elements of the MIMO radar is M=7 and N=15 respectively, the spacing between the transmitting and receiving array elements is half of the signal wavelength, each transmitting array transmits mutually orthogonal Hadamard coded signals, the number of codes in each pulse repetition period is 256, and there are three far-field incoherent targets. The root mean square error of angle estimation is defined as Among them, P is the number of targets; M T is the number of Monte Carlo experiments; For the p-th target at the m-th t DOD estimation in the Monte Carlo experiment; For the p-th target at the m-th t The DOA estimate in the Monte Carlo experiment; the signal-to-noise ratio is defined as In the following simulation experiments, M T =100. In the method of the present invention, ρ=1.1, regularization parameter λ=20, β0=10 -1 , when the iteration condition is satisfied Or when the maximum number of iterations k=1000 is reached, the iteration stops, where Represents the estimated value of the covariance tensor at the kth iteration.

[0171] Simulation Experiment 1: Target Angle Estimation Constellation Diagrams of Different Methods Under Array Element Failure

[0172] In this experiment, the signal-to-noise ratio SNR is set to -5dB, the number of snapshots Q is set to 100, and the number of Monte Carlo experiments is set to M. T =100, the asterisk indicates the estimated angle value, the cross indicates the true angle value, the 2nd and 5th elements in the transmitting array are invalid, and the 1st, 4th, 7th, 12th, and 14th elements in the receiving array are invalid. Figure 2 (a) It can be seen that the ESPRIT algorithm fails without compensating for the failed array element. The method of the present invention and the second prior art both use the multi-dimensional structure of array data to recover the missing data of the failed array element, such as Figure 2 As shown in (b), for most Monte Carlo experiments, the second prior art can correctly estimate the target angle, but some experiments produce incorrect angle estimation results. This is because the second prior art only considers the Vandermonde structure of the factor matrix when reconstructing the missing data of the failed array element, but ignores the relationship between the factor matrices, resulting in instability when reconstructing the missing data. Figure 2 (c) It can be seen that the method of the present invention can accurately estimate the target angle in all Monte Carlo experiments, which is mainly due to the full use of the Vandermonde structure of the factor matrix and the relationship between the factor matrices in tensor filling.

[0173] Simulation Experiment 2: Convergence Analysis of the Method of the Present Invention and the Prior Art 2

[0174] This experiment analyzes the convergence of the method of the present invention and the second prior art. Figure 3 The curve diagram of the relationship between the convergence rate and the number of iterations, where the simulation parameters are the same as those of simulation experiment 1. The vertical axis is the residual value Expressed in logarithmic form. Figure 3 As can be seen, the relative change in the residual value of the method of the present invention decreases more rapidly with increasing iterations, leading to a faster convergence rate. Conventional technology 2 requires more iterations to achieve convergence. Conventional technology 2 randomly initializes the factor matrix, resulting in a slow convergence rate. However, the method of the present invention uses the factor matrix obtained by performing CP decomposition on the covariance tensor of the MIMO radar under pixel failure as the initialization value, accelerating convergence and reducing reconstruction error.

[0175] Simulation Experiment 3: Relationship between target angle estimation error and signal-to-noise ratio of different methods

[0176] In this experiment, the signal-to-noise ratio range is set to -30 to 0dB, and the other simulation parameters are the same as those in simulation experiment 1. Figure 4It can be seen that since the array element failure destroys the integrity of the covariance matrix structure, the angle estimation accuracy is poor when the ESPRIT algorithm is directly used under array element failure, that is, the angle of the target cannot be effectively estimated; since the missing data of the failed array element is restored, the prior art one and the prior art two and the method of the present invention have good estimation performance, especially in the low signal-to-noise ratio area. The prior art one and the prior art two and the method of the present invention have almost the same performance as the ESPRIT algorithm under normal circumstances (that is, when there is no array element failure), but overall, the performance of the method of the present invention is significantly better than that of the prior art one and the prior art two, especially under high signal-to-noise ratio conditions.

[0177] Simulation Experiment 4: Variation of target angle estimation error with snapshot number for different methods

[0178] In this experiment, the number of snapshots is set to vary from 50 to 350, the signal-to-noise ratio is -5dB, and the rest of the simulation parameters are the same as those in simulation experiment 1. Figure 5 As can be seen, the angle estimation accuracy of all methods improves with the increase in the number of snapshots, but the method of the present invention significantly outperforms the matrix-filling-based method (Existing Technique 1). This demonstrates that the multidimensional structure of MIMO radar array data plays a crucial role in improving the accuracy of missing data recovery due to element failures. Furthermore, since tensor filling imposes constraints between factor matrices, the stability of missing data recovery is improved, resulting in a method of the present invention that achieves better angle estimation performance than Existing Technique 2.

[0179] Simulation Experiment 5: Relationship between the Angle Estimation Error of Different Methods and the Number of Failed Array Elements

[0180] In this experiment, two array element failure scenarios are considered:

[0181] Case 1: The fourth element of the transmitting array fails. The number of failed elements in the receiving array increases from 1 to 8, and the position of each failed element changes randomly.

[0182] Case 2: The second and fifth elements of the transmitting array fail. The number of failed elements in the receiving array increases from 1 to 8, and the positions of the failed elements change randomly each time.

[0183] In the above two cases, the relationship between the angle estimation performance and the number of failed receiving elements is as follows: Figure 6 As shown in Figure 7, where the signal-to-noise ratio SNR = -5dB and the number of snapshots Q = 100. 100 experiments were conducted for each case. Figure 6 and Figure 7It can be seen that as the number of failed receiving elements increases, the angle estimation performance of each method deteriorates to varying degrees, but the method of the present invention has the best angle estimation accuracy in both cases.

[0184] Simulation Experiment 6: Comparison of Running Time of Different Methods

[0185] Table 1 Running time of different DOA estimation methods

[0186]

[0187] The simulation settings for this experiment were the same as those for Simulation Experiment 1, using MATLAB 2018a, an Intel Core i5-4570 CPU, and 8GB of memory. As shown in Table 1, compared to the first and second prior arts, the proposed method has a shorter runtime and better angle estimation performance.

Claims

1. A method for estimating the angle of an element-failed MIMO radar based on a factor matrix prior, characterized in that: The following steps are involved: (1) Matched filtering is performed on the received signal of the bistatic MIMO radar with array element failure to obtain the output signal of the virtual array And calculate and obtain the virtual array covariance matrix R; (2) Using the virtual array covariance matrix Constructing a fourth-order covariance tensor The covariance tensor Expressed as a tensor CANDECOMP / PARAFAC decomposition model; specifically: (2.1) Dual-base MIMO radar virtual array covariance matrix Can be rearranged into a fourth-order covariance tensor in,(·) * indicates conjugation; Represents Tucker operation; (2.2) Re-adjust the factor matrix Make all elements in g equal to 1. Then eliminate g to get the simplified expression of the covariance tensor (2.3) Due to the perturbation of the permutation and scaling effects of the CP decomposition, the covariance tensor The factor matrix is expressed as and in and represents four factor matrices; represents the permutation matrix; Λ1, Λ2, Λ3, and Λ4 are P×P real-valued diagonal matrices, and the diagonal elements correspond to the scaling coefficients; N1, N2, N3, and N4 represent the fitting errors; (2.4) Since the transmit and receive steering matrices A of the MIMO radar t 、A r are all Vandermonde structures, and the factor matrix and Each column vector of A is a different constant t and A r The corresponding column vectors of are scaled respectively, so when the fitting error term is ignored, the matrix and It also has a Vandermonde structure, that is, the factor matrix U (n) ,n=1,…,4, with Vandermonde structure; (2.5) Ignoring the influence of the fitting error term, the relationship between the factor matrices is: and The diagonal matrix Diagonal matrix (3) By utilizing the Vandermonde structure characteristics of the factor matrix, the mutual relationship between the factor matrices, and the low CP rank of the tensor, a tensor filling model with prior constraints on the factor matrix is established; specifically: In order to effectively recover the missing data of failed array elements, a tensor filling model for recovering missing data of failed array elements is established by utilizing the Vandermonde structure characteristics of the factor matrix, the mutual relationship between the factor matrices, and the low CP rank of the tensor: Among them, M (n) ,n=1,2,3,4, represents the auxiliary matrix, Ω represents the incomplete covariance tensor The set of known non-zero elements in ; represents the projection operator onto the set Ω; ||·|| * represents the nuclear norm; ||·|| F represents the Frobenius norm; λ represents the regularization parameter; Represents a block Hankel matrix operation, defined as the matrix Transform to block Hankel matrix operation, where u p represents the p-th column of U, p=1,2,…,P; S1+S2=S+1; Represents the Hankel operation, defined as the vector Transformed into Hankel matrix operation, where S1+S2=S+1; (4) To solve the constrained tensor filling model, the tensor filling model in step (3) is converted into an unconstrained augmented Lagrangian function form; (5) Use the ADMM algorithm to iteratively solve the augmented Lagrangian function in step (4), and obtain the factor matrix U at the end of the iteration (1) ,U (2) and the diagonal matrix Δ (3) ,Δ (4) ; (6) According to the factor matrix U (1) ,U (2) and the diagonal matrix Δ (3) ,Δ (4) , using Tucker operations to construct the complete covariance tensor Then it is restored to the complete covariance matrix R through symmetric Hermitian expansion, and finally the ESPRIT algorithm is used to estimate the target angle.

2. The method for estimating the angle of an element failure MIMO radar based on a factor matrix prior according to claim 1, characterized in that: The step (1) is specifically as follows: (1.1) Both the transmit array and receive array of a bistatic MIMO radar are uniform linear arrays, consisting of M transmit elements and N receive elements, respectively. When an element fails, the echo signal collected by the bistatic MIMO radar is subjected to matched filtering. After pulse accumulation, the resulting echo signal matrix output by MN virtual elements is: in, is the virtual array echo signal matrix under Q snapshots; and are the manifold matrices of the transmitting array and receiving array when there are failed array elements; is the Gaussian white noise matrix; S∈ P×Q is the target coefficient matrix, where P is the independent far-field target; ⊙ represents the Khatri-Rao product; (1.2)Ω T and Ω R They are the position sets of failed transmitting array elements and failed receiving array elements respectively. When the pth t ∈Ω T When an array element fails, its manifold matrix The first p t The rows are all zero; when the pth row in the receiving array r ∈Ω R When an array element fails, the manifold matrix The first p r All rows are zero; the maximum likelihood estimate of the virtual array covariance matrix under the Q pulse period is: in,(·) H represents the conjugate transpose, R s =SS H / Q = diag(ρ) represents the signal covariance matrix, diag(ρ) represents the diagonal matrix generated by the vector ρ, ρ = [ρ1, ρ2, ..., ρ P ],ρ p ,p=1,2,…,P, represents the reflection coefficient of the pth target, represents the noise covariance matrix.

3. The method for estimating the angle of an element failure MIMO radar based on a factor matrix prior according to claim 1, characterized in that: The step (4) is specifically as follows: The optimization model in step (3) is expressed as an unconstrained augmented Lagrangian function: For simplicity, we define ={Δ (3) ,Δ (4) }、 and β>0 indicates penalty coefficient; D (n) represents the Lagrange multiplier matrix, n=1, 2, 3, 4; <·> represents the inner product.

4. The method for estimating the angle of an element failure MIMO radar based on a factor matrix prior according to claim 1, wherein: The step (5) is specifically as follows: The optimization problem in step (4) is solved iteratively using the alternating direction multiplier algorithm, where the variables are updated in the kth iteration and The steps are as follows: (5.1) In the kth iteration, by solving the optimization problem To update the variable Transform the subproblem into solving U (1) and U (2) Convex optimization problem: Where, (·) T represents transpose; R (n) and Tensors and The solution of the convex optimization problem (3) satisfies: in, express The inverse transform of U (n) The closed-form solution of each row is used to obtain U (n) The solution of and W i =diag(Ω (n) (i,:)), i∈{1,2,…,I n }; when n = 1, I1 = M; when n = 2, I2 = N; then formula (4) can be transformed into: λU (n) (i,:)P k (n) W i P k (n)H +β k U (n) (i,:)=C(i,:)(5) Where U (n) (i,:) represents the matrix U (n) The i-th row; C(i,:) represents the i-th row in the matrix C, so the row vector U (n) The closed-form solution of (i,:) is expressed as: AT (n) (i,:)=C(i,:)[λP k (n) IN i P k (n)H +β k AND] -1 (6) Then, by solving U (n) Each line Get the factor matrix U (n) =[(U (n) (i,:)) T L (U (n) (I n ,:)) T ] T , n=1,2; (5.2) Regarding the solution of Δ={Δ (3) ,Δ (4) The convex optimization problem of} is expressed as: Similarly, the solution to this convex optimization problem satisfies: Formula (8) can be further expressed as: Where, To solve the diagonal matrix of the i-th system of equations, n=3,4,i=1,2,…,I n And when n=3, I3=M, when n=4, I4=N; therefore Δ i (n) The closed-form solution can be expressed as: in, Represents the generalized inverse; To reduce the estimation error, Δ i (n) Take the average as Δ (n) Solution: (5.3) About variables The optimization problem is expressed as: Using the SVT algorithm to solve the matrix nuclear norm minimization problem, the solution to the above optimization problem is: in, is the singular value soft threshold shrinkage operator, 1 / β k is the threshold; (5.4) Variables The update expression is: (5.5) The expression for updating β is β k+1 =ρβ k .

5. The method for estimating the angle of MIMO radar with element failure based on factor matrix prior according to claim 1, characterized in that: The step (6) is specifically as follows: The factor matrix U obtained according to step (5) (1) ,U (2) and the diagonal matrix Δ( 3 ),Δ( 4 ), using Tucker operation to construct the complete covariance tensor and will The covariance matrix is expanded into a symmetric Hermitian matrix, and the target angle is finally estimated using the ESPRIT algorithm.

6. A computer storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the method for estimating the angle of an element failure MIMO radar based on factor matrix prior according to any one of claims 1 to 5 is implemented.

7. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the computer program, the method for estimating the angle of an element failure MIMO radar based on factor matrix prior according to any one of claims 1 to 5 is implemented.

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